Learn Swedish: Jag känner hans fru.

No, it doesn't! That is one of the nice things about third-person possessive pronouns like hans (his), hennes (her), and deras (their). They never change form, no matter if the noun is an en-word, an ett-word, or plural. It is always hans fru, hans barn, and hans bilar.

Why is the word for "know" känner here instead of vet?

In Swedish, you use känner when you are talking about knowing a person, a place, or being familiar with something. You use vet when you know a fact or a piece of information. Since you are talking about knowing a person (his wife), you must use känner.

Is the 's' at the end of hans related to the English apostrophe 's'?

Exactly! Just like in English where "John's" shows possession, Swedish adds an s to names and nouns to show ownership. The word hans literally comes from adding that possessive s to the pronoun han (he). Just remember that Swedish does not use an apostrophe before the s.

How would I change this sentence to say "I know her husband"?

You would say Jag känner hennes man. The word hennes is the third-person possessive pronoun for "her". Just like hans, it stays exactly the same no matter what noun comes after it.